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On a new extension of BTP for binary CSPs

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Abstract

The study of broken-triangles is becoming increasingly ambitious, by both solving constraint satisfaction problems (CSPs) in polynomial time and reducing search space size through either value merging or variable elimination. Considerable progress has been made in extending this important concept, such as dual broken-triangle and weakly broken-triangle, in order to maximize the number of captured tractable CSP instances and/or the number of merged values. Specifically, m-wBTP allows us to merge more values than BTP. DBTP, ∀∃-BTP, k-BTP, WBTP and m-wBTP permit us to capture more tractable instances than BTP. However, except BTP, none of these extensions allows variable elimination while preserving satisfiability. Moreover, k-BTP and m-wBTP define bigger tractable classes around BTP but both of them generally need a high level of consistency. Here, we introduce a new weaker form of BTP, called m-fBTP for flexible broken-triangle property, which will represent a compromise between most of these previous tractable properties based on BTP. m-fBTP allows us on the one hand to eliminate more variables than BTP while preserving satisfiability and on the other to define a new bigger tractable class for which arc consistency is a decision procedure. Likewise, m-fBTP permits to merge more values than BTP but fewer than m-wBTP. The binary CSP instances satisfying m-fBTP are solved by algorithms of the state-of-the-art like MAC and RFL in polynomial time. An open question is whether it is possible to compute, in polynomial time, the existence of some variable ordering for which a given instance satisfies m-fBTP.

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Notes

  1. Given a binary CSP instance \(I=(X,C)\), the micro-structure of I is the undirected graph \(\mu (I) =(V,E)\) with:

    • \(V = \{ (x_{i},v_{i}): x_{i} \in X, v_{i} \in D(x_{i}) \}\),

    • \(E = \{\ \{(x_{i},v_{i}),(x_{j},v_{j})\}\ \mid \ i\neq j,C_{ij} \notin C\text { or }C_{ij} \in C,(v_{i},v_{j}) \in Rel(C_{ij}) \}\)

  2. A complete subgraph where each pair of vertices are connected.

  3. A class \({\Gamma } \) of CSP instances is said to be conservative with respect to a filtering consistency \(\phi \) if it is closed under \(\phi \), that is, if the instance obtained after the application of \(\phi \) still belongs to \({\Gamma }\).

  4. Given a binary CSP instance \(I=(X,C)\), the Micro-structure based on Dual of I is the undirected graph \((V,E)\) such that:

    • \(V = \{ (C_{i},t_{i}): C_{i} \in C, t_{i} \in Rel(C_{i}) \}\),

    • \(E = \{\ \{(C_{i},t_{i}),(C_{j},t_{j})\}\ \mid \ i\neq j, t_{i} [ Scp(C_{i}) \cap Scp(C_{j}) ] = t_{j} [ Scp(C_{i}) \cap Scp(C_{j}) ] \}\)

    where \(t_{k}[Y]\) denotes the restriction of \(t_{k}\) to the variables in Y.

  5. A binary CSP instance I satisfies i-consistency if any consistent assignment to \((i - 1)\) variables can be extended to a consistent assignment on any \(i^{th}\) variable. A binary CSP instance I satisfies strong k-consistency if it satisfies i-consistency for all i such that \(1 < i \leq k\).

  6. http://www.cril.univ-artois.fr/CPAI08

References

  1. Montanari, U. (1974). Networks of constraints: fundamental properties and applications to picture processing. Artificial Intelligence, 7, 95–132.

    MathSciNet  MATH  Google Scholar 

  2. Mackworth, A.K. (1977). Consistency in networks of relations. Artificial Intelligence, 8, 99–118.

    Article  MATH  Google Scholar 

  3. Freuder, E.C. (1991). Eliminating interchangeable values in constraint satisfaction problems. In Proceedings of the 9th national conference on artificial intelligence, (Vol. 1 pp. 227–233).

  4. Likitvivatanavong, C., & Yap, R.H.C. (2013). Many-to-many interchangeable sets of values in csps. In Proceedings of SAC (pp. 86–91).

  5. Cooper, M.C., El Mouelhi, A., Terrioux, C., Zanuttini, B. (2014). On Broken Triangles. In Principles and practice of constraint programming - 20th international conference, CP 2014. Proceedings (pp. 9–24).

  6. Cooper, M.C., Duchein, A., El Mouelhi, A., Escamocher, G., Terrioux, C., Zanuttini, B. (2016). Broken triangles: from value merging to a tractable class of general-arity constraint satisfaction problems. Artificial Intelligence, 234, 196–218.

    Article  MathSciNet  MATH  Google Scholar 

  7. Cohen, D.A., Cooper, M.C., Escamocher, G., Zivny, S. (2013). Variable elimination in binary CSP via forbidden patterns. In IJCAI 2013, Proceedings of the 23rd international joint conference on artificial intelligence (pp. 517–523).

  8. Cohen, D.A., Cooper, M.C., Escamocher, G., Zivny, S. (2015). Variable and value elimination in binary constraint satisfaction via forbidden patterns. Journal of Computer and System Sciences, 81(7), 1127–1143.

    Article  MathSciNet  MATH  Google Scholar 

  9. Dechter, A., & Dechter, R. (1987). Removing redundancies in constraint networks. In Proceedings of the 6th national conference on artificial intelligence (pp. 105–109).

  10. Cooper, M.C., Jeavons, P., Salamon, A. (2008). Hybrid tractable CSPs which generalize tree structure. In Proceedings of ECAI (pp. 530–534).

  11. Cooper, M.C., Jeavons, P., Salamon, A. (2010). Generalizing constraint satisfaction on trees: hybrid tractability and variable elimination. Artificial Intelligence, 174, 570–584.

    Article  MathSciNet  MATH  Google Scholar 

  12. Naanaa, W. (2013). Unifying and extending hybrid tractable classes of csps. Journal of Experimental & Theoretical Artificial Intelligence, 25(4), 407–424.

    Article  Google Scholar 

  13. Cooper, M.C. (2014). Beyond consistency and substitutability. In Principles and practice of constraint programming - 20th international conference, CP 2014. Proceedings (pp. 256–271).

  14. El Mouelhi, A., Jégou, P., Terrioux, C. (2014). Hidden tractable classes: from theory to practice. In 26th IEEE international conference on tools with artificial intelligence, ICTAI, 2014 (pp. 437–445).

  15. El Mouelhi, A., Jégou, P., Terrioux, C. (2015). A hybrid tractable class for non-binary CSPs. Constraints, 20(4), 383–413.

    Article  MathSciNet  MATH  Google Scholar 

  16. Jégou, P., & Terrioux, C. (2015). The extendable-triple property: A new CSP tractable class beyond BTP. In Proceedings of AAAI (pp. 3746–3754).

  17. Cooper, M.C., Jégou, P., Terrioux, C. (2015). A microstructure-based family of tractable classes for CSPs. In Principles and practice of constraint programming - 21st international conference, CP, 2015, Proceedings (pp. 74–88).

  18. Naanaa, W. (2016). Extending the broken triangle property tractable class of binary csps. In Proceedings of the 9th hellenic conference on artificial intelligence, SETN, 2016 (pp. 3:1–3:6).

  19. Cooper, M.C., El Mouelhi, A., Terrioux, C. (2016). Extending broken triangles and enhanced value-merging. In Principles and practice of constraint programming - 22nd international conference, CP 2016, Proceedings (pp. 173–188).

  20. Sabin, D., & Freuder, E.C. (1994). Contradicting Conventional Wisdom in Constraint Satisfaction. In Proceedings of ECAI (pp. 125–129).

  21. Nadel, B. (1988). Tree search and arc consistency in constraint-satisfaction algorithms. In Search in artificial intelligence (pp. 287–342). Berlin: Springer.

  22. El Mouelhi, A. (2017). A btp-based family of variable elimination rules for binary csps. In To appear inproceedings of the thirty-first AAAI conference on artificial intelligence.

  23. Dechter, R., & Pearl, J. (1987). The Cycle-cutset method for improving search performance in AI Applications. In Proceedings of the third IEEE on artificial intelligence applications (pp. 224–230).

  24. Rossi, F., Petrie, C.J., Dhar, V. (1990). On the equivalence of constraint satisfaction problems. In ECAI (pp. 550–556).

  25. Jégou, P. (1993). Decomposition of domains based on the micro-structure of finite constraint satisfaction problems. In Proceedings of the 11th national conference on artificial intelligence, 1993 (pp. 731–736).

  26. El Mouelhi, A., Jégou, P., Terrioux, C., Zanuttini, B. (2013). Some new tractable classes of csps and their relations with backtracking algorithms. In 10th international conference integration of AI and OR techniques in constraint programming for combinatorial optimization problems, CPAIOR 2013, May 18-22, 2013. Proceedings (pp. 61–76).

  27. Jeavons, P., & Cooper, M. (1995). Tractable constraints on ordered domains. Artificial Intelligence, 79(2), 327–339.

    Article  MathSciNet  MATH  Google Scholar 

  28. Dechter, R., & Pearl, J. (1989). Tree-Clustering For constraint networks. Artificial Intelligence, 38, 353–366.

    Article  MathSciNet  MATH  Google Scholar 

  29. El Mouelhi, A., Jégou, P., Terrioux, C. (2013). A hybrid tractable class for non-binary CSPs. In 2013 IEEE 25th international conference on tools with artificial intelligence, November 4-6, 2013 (pp. 947–954).

  30. El Mouelhi, A., Jégou, P., Terrioux, C. (2013). Microstructures for csps with constraints of arbitrary arity. In Proceedings of the Tenth symposium on abstraction, reformulation, and approximation, SARA 2013, 11-12 July 2013.

  31. Freuder, E.C. (1982). A sufficient condition for Backtrack-Free search. JACM, 29(1), 24–32.

    Article  MathSciNet  MATH  Google Scholar 

  32. El Mouelhi, A. (2014). Classes polynomiales pour CSP : De la théorie à la pratique. Ph.D. thesis, Aix-Marseille Université.

  33. Jégou, P. (1993). On the consistency of general constraint-satisfaction problems. In Proceedings of the 11th national conference on artificial intelligence. July 11-15, 1993 (pp. 114–119).

  34. Debruyne, R., & Bessière, C. (2001). Domain filtering consistencies. Journal of Artificial Intelligence Research, 14, 205–230.

    Article  MathSciNet  MATH  Google Scholar 

  35. Cooper, M.C., & Zivny, S. (2016). The power of arc consistency for CSPs defined by partially-ordered forbidden patterns. In Proceedings of the 31st annual ACM/IEEE symposium on logic in computer science, LICS ’16, 2016 (pp. 652–661).

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    Achref El Mouelhi

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El Mouelhi, A. On a new extension of BTP for binary CSPs. Constraints 23, 355–382 (2018). https://doi.org/10.1007/s10601-018-9290-9

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